Long-time stochasticity for periodic iterations of random surface maps

Abstract

Let $M$ be a closed Riemannian surface, and let $f_1,\ldots,f_s$ be area-preserving $C^2$ diffeomorphisms with fixed positive probabilities such that the resulting random dynamical system is uniformly expanding on average. For each $n$, choose $ω_1,\ldots,ω_n$ independently according to these probabilities and set $F=f_{ω_n}\circ\cdots\circ f_{ω_1}$. We prove that there exists $α>0$ such that, with probability at least $1-e^{-αn}$, the correlations of any two Hölder observables $ϕ,ψ$ under $F^m$ are bounded by $e^{-αn}\|ϕ\|_{C^β}\|ψ\|_{C^β}$, for all $m\leq e^{αn}$. These estimates imply a law of large numbers and a central limit theorem for the Birkhoff sums along $F$ of any zero-mean Hölder observable $ϕ$, for exponentially long times in $n$. The limiting Gaussian has variance $\int_Mϕ^2\,d\mathrm{vol}$.